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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-1uplex | Structured version Visualization version GIF version |
Description: A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-1uplex | ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-pr11val 34320 | . . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
2 | bj-pr1ex 34321 | . . 3 ⊢ (⦅𝐴⦆ ∈ V → pr1 ⦅𝐴⦆ ∈ V) | |
3 | 1, 2 | eqeltrrid 2918 | . 2 ⊢ (⦅𝐴⦆ ∈ V → 𝐴 ∈ V) |
4 | df-bj-1upl 34313 | . . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
5 | p0ex 5285 | . . . 4 ⊢ {∅} ∈ V | |
6 | bj-xtagex 34304 | . . . 4 ⊢ ({∅} ∈ V → (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V)) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V) |
8 | 4, 7 | eqeltrid 2917 | . 2 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V) |
9 | 3, 8 | impbii 211 | 1 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∈ wcel 2114 Vcvv 3494 ∅c0 4291 {csn 4567 × cxp 5553 tag bj-ctag 34289 ⦅bj-c1upl 34312 pr1 bj-cpr1 34315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-fal 1550 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-xp 5561 df-rel 5562 df-cnv 5563 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-bj-sngl 34281 df-bj-tag 34290 df-bj-proj 34306 df-bj-1upl 34313 df-bj-pr1 34316 |
This theorem is referenced by: bj-2uplex 34337 |
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